Images, posts & videos related to "Rational Number"

How do you turn 8.2 into a rational number?

DAE find it pleasant (or even necessary!!!) to make the gas pump stop at a nice even price/number? I don't even have a good rational reason other than, because $40.00 is nicer to look at than $39.96. Why are we like that???

I've done this for so long, that I don't even remember not doing it! My kid asked why I don't just take the handle out when it stops pumping. I don't even have a good rational answer other than, because $40.00 is nicer to look at than $39.96.

[Algebra] rational numbers that are not integers

Prove that every positive rational number x can be expressed in one and only one way in the form x = a1+a2/1.2 +a3/1.2.3+....+ak/1.2.3..k. where 0<=a1; 0<=a2<2,......0<=ak<k. I recently took inequalities in my analysis class and I was assigned this as homework.

Can someone plz help me on how to start?The inequalities i took are Cauchy shwartz,y>0 so y+1/y>=2 AGM,(1+x)^n >= 1+nx s.t x>=-1 and b^n - a^n < n(b-a)b^n-1. I also took induction. Thanks in advance.

Okay real quick. Even when we do see the numbers. Lets all be rational and decide what to do next

Rational non-whole number square roots

Are there any square roots that are rational numbers (other than the whole numbers)?

If you add every real rational number together, you just get zero

Is there an induction method to prove for all rational numbers?

We have the classic induction to prove for all n bigger that an arbitrary number, which is to prove `P(n)βP(n+1)`

, and as much as I can remember Cauchy Induction (aka Forward-Backward Induction) which is `P(n)βP(2n)`

and `P(n)βP(n-1)`

which also proves for all n β β€. But do we have a method to prove for all rational numbers? I was going to ask whether it's possible to prove for all n β β but then thought it was impossible because we can't represent transcendental numbers such as e or Ο algebraically.

Holy fucking shit I hate rational numbers

Those little fucks are anything but rational. Fuck them, why does one thing do this and another similar thing do something completely different? What the fuck there are exponents and variables now??? Holy fuck why. Why does that squiggly line hate me so much why doesn't this make any sense I've been doing this for a week and I still feel like a juvenile chimpanzee who was given access to a high caliber computer and told to write code for a Projekt Red game. Holy fuck I'm going to drop out of college. I'm already juggling like three different assignments and this shit really just needs to make my life worse huh? I've been in a perpetual state of panic since Monday. Oh my god I hate it here

New C library for weird number systems such as base -1+i, base -2, Stern-Brocot rationals, base Fibonacci github.com/natelastname/hβ¦

Are there any exotic CPUs that implements rational number arithmetic?

*implement

Does every rational number with a terminating decimal representation also has an alternative representation as a repeating decimal?

The question may seem trivial but in fact it is not. Are there rational numbers that have only one possible representation (as a terminating decimal)? If yes, what is the (real) reason for their existence? Can those restrictions be overcome?

I'm trying to answer if every fraction is a rational number. Can you have an irrational number in your numerator or denominator and still call it a fraction? For example, would pi/2 be considered a fraction? Would 2/square root of 2 still be considered a fraction?

Despite record high case numbers, Ontarioβs public health officials supported the reopening of indoor dining, bars, gyms and cinemas. Here was the rational of Ottawaβs top doctor. ottawa.ctvnews.ca/read-thβ¦

The internet as it exists today is the number one contributor to mass social decline, decline in rational and critical thinking, and decline in common sense and intelligence

Tbh i don't even really know if this qualifies as an unpopular opinion anymore as i think a lot of people recognize this in some capacity, but the internet is largely the cause of the sharp and steady decline of culture, social norms and rational thinking

there are, clearly and obviously, positive sides to having the internet available to society at large. It's cool to have access to online learning and educational platforms, email, weather, GPS, etc. and having online tools for group work and collaboration

But the downfall of having everything online far outweighs the positives. Social media is evil, it has given every narcissist and idiot a platform to spread superficiality and hate. Misinformation and pseudo science is everywhere. No one wants to hear opinions that contrast against their own. Hate is everywhere you look. Its little more than a sad depressing downward spiral that many of us are powerless to stop or control. Politics has become a cesspool. Twitter is absolute and utter garbage to look at. i could go on and on

I just think about this pandemic and how differently things could've gone if the internet never existed. We as a society would be infinitely better off relying on traditional sources of information and media. All the echo chambers wouldn't exist.

And yes I realize I'm on the internet disparaging the internet, which feels totally ironic and futile. Part of me just wishes that I could wake up in a world without internet tomorrow, everything else remains the same, and we would all be so much better off for it

Help please , could someone explain rational and irrational numbers ? The way they are explaining it is just confusing they say examples of rational numbers are fractions but then say they canβt be ?? Thanks

Let f(x) = 0 of x is any rational number and f(x) = 1 if x is any irrational number. Show that f is not integrable on [0,1].

Could an answer to this be since there is an infinite number of rational and irrational numbers on any interval, the function would have an infinite number of jump discontinuities and is therefore non integrable?

Find rational numbers a,b for which (4aΒ³ - bΒ³) / (12b) is a square number

**Edit: Forgot an important detail: a not equal to b**

**Edit2: In the title I wrote "is a square number", but I actually meant "is the square of a rational number"**

Hey! I was working on the generalization of problem, which resulted in the quadratic equation

xΒ² - bx - (aΒ³ - bΒ³)/(3b) = 0

for which the solutions are of the form

-b/2 +- sqrt((4aΒ³ - bΒ³)/(12b))

and I wanted to see if there are any "nice" values for a and b ("nice" meaning rational in this case) with a not equal to b such that the quadratic equation has rational solutions. I couldn't find any, so I'm not even sure if there are any.

Is it possible to make a 2D physics simulation engine based on rational numbers?

Just a thought that crossed my mind. Most (all?) physics simulation engines are not precise, because they use floating-point numbers. If we used rational numbers (with arbitrary precision) instead, we could, at least theoretically, achieve complete precision.

In such an engine, all positions would be rational numbers, all angles would be rational multiples of Pi and time would proceed on a rational timeline.

The engine would simulate simple, rigid-body Newtonian physics.

Would it be possible to make such an engine, or is there something in Newtonian physics that would be an obstacle to this?

EDIT: Really curious that the comment which most misunderstood the question is getting most upvotes.

> Floating-point numbers are already rational numbers.

Yeah, but rational numbers are not all floating-point. There are many rational numbers which are not expressible as floating-point (1/10 for example). Thatβs all that my question is about. Could we build a physics engine on actual rational numbers?

However, thanks for all the other actually insightful answers, so far!

The 'carat' of a rational number represents just how 'golden' a ratio is!

A beautiful result in number theory is that the Euclidean algorithm for computing greatest common divisor has a worst case computational complexity at consecutive Fibonacci numbers. Related is the fact that the number of steps in the algorithm on the inputs (x,y) generally increases as the fraction x/y becomes a good approximation of the golden ratio, which is approximately 1.61803...

For example, let's do the Euclidean algorithm (x,y) = (161, 100) to represent the approximation 161/100 = 1.61:

161 = 1(100) + 61

100 = 1(61) + 39

61 = 1(39) + 22

39 = 1(22) + 17

22 = 1(17) + 5

17 = 3(5) + 2

5 = 2(2) + 1;

gcd(161, 100) = 1

As we can see, each line in the form a = q(b) + r has q = 1 for the first 5 lines, 17/5 was the first ratio >2. We then say that the ratio 1.61 is 5 carat gold. Compare this with the Euclidean algorithm for (21,13), much smaller inputs, but these are consecutive Fibonacci numbers (1,1,2,3,5,8,13,21). It is an easy exercise to check that the q terms are always 1, and the r terms descend down the Fibonacci sequence at each step of the algorithm, (e.g. 21 = 1(13) + 8) and so the ratio 21/13 = 1.615... is also 5 carat gold. Generally, if F_n is the nth Fibonacci number than the ratio F_(n+1) / F_n is n carat gold.

I'm not the first to have this idea, but I might be the first silly enough to give it this name. Carat can obviously exceed 24, which makes it especially silly. The carat of a ratio can also be put in terms of continued fractions, which gives a nice extension to all real numbers, where the carat of the true golden ratio is infinity.

prove that there is a permutation of n rational numbers that the average of two numbers is not between them

for example:

{1,2,3,4}

This permutation satisfies this condition: 1,3,2,4

but this permutation doesn't satisfy the condition: 4,3,2,1(because between 3 and 1(the average is 2) there is 2)

prove that using induction.

any hint?

edit: **n different rational numbers***

Need help with a proof for rational and irrational numbers

An exercise from my Discrete Maths problem set asks: "Prove that, if a and b are rational numbers with a =/= b, then a + (1/sqrt(2)) * (b - a) is irrational."

Is it sufficient to reference a previous proof showing that 1/sqrt(2) is irrational, then say that, since b - a =/= 0, this means that (1/sqrt(2)) * (b - a) must be irrational? All other irrational number proofs I've seen have been proofs by contradiction, but I'm unsure how to arrive at a contradiction here.

Why does the diagonalization argument prove Irrational numbers uncountable, can't you do it to rationals?

I understand the diagonalization argument on why the Irrational numbers are uncountable (Image down below) but my central confusion is couldn't you do the same thing to the rational numbers between 0-1 and build one that's, not on the list, but I know the rational numbers are countable so how would that show irrationals are uncountable. That is my central confusion if you could explain to me that would be amazing, Thank you!

https://preview.redd.it/a8m9f0wfqyv51.png?width=1142&format=png&auto=webp&s=cbbbebfd653e8734f8fe40a549d3f4250ebb33e5

All 4 of the answers are rational numbers. My pre calc teacher refuses to believe 22/7 is rational

Sums of regular sets of rational numbers

Consider the set S of strings matching the regular expression `[1-9][0-9]*/[1-9][0-9]*`

. That is, any string in this set consists of a digit in {1,2,...9}, followed by any number of digits in {0,1,2,...9}, followed by a `/`

character, followed by a digit in {1,2,...9}, followed by any number of digits in {0,1,2,...9}. For example, such strings include `2/4`

and `17/1`

and `134/273`

. There is a natural function from S to the set of positive rational numbers Q that results from interpreting the string in S as a fraction. Call this function F, and observe that `F("2/4") = 1/2`

for example. Similarly, let G be the function that maps from Q to S which, given a positive rational number q, writes q in its lowest form (the canonical form) as a fraction and outputs the fraction as a string in S. For example, `G(1/2) = "1/2"`

and `G(3) = "3/1"`

.

Can you describe an algorithm that, given a regular expression R, determines whether the sum over all strings s in S that match R of F(s) converges? (For example, if the regular expression was `R = 1/[1-9][0-9]*`

then your algorithm should output `DIVERGES`

because this sum corresponds to the harmonic series, which diverges. But conversely, if the regular expression was `R = 1/10*`

your algorithm should output `CONVERGES`

because this corresponds to the sum 1 + 1/10 + 1/100 + 1/1000 ... which converges.)

*(A harder version of the problem.)* Can you describe an algorithm that, given a regular expression R, determines whether the sum of q over all positive rational numbers q in Q such that G(q) matches R converges?

Marine: "It is widely believed that science can provide rational explanations for the countless phenomena of our universe. However, there are many aspects of our existence that science has not yet found a solution to and cannot decipher with numbers."

If you choose a random real number, what is the probability that it is rational?

I'm guessing this question doesn't have a well defined answer. If it does, what is it? If it doesn't, why not?

The most irrational looking rational number?

There is a nice proof that at least one of e + pi or e*pi is irrational, but neither one is known to be so. Still, looking at these numbers one's gut instinct is that they should be irrational (if not transcendental). This got me wondering- are there any really nice examples of numbers that "look irrational (or transcendental)" but are known to be rational? I know "look irrational" isn't precise, so there is some freedom in interpretation here. My loose working definition is "I would bet $1000 that it is irrational."

Anyways, I thought this might generate some interesting examples of rational numbers!

Ever think you died and are now in an alternate reality? After an injury felt that everything is off? Seen things that defy physics? License plates numbers in different places/times mathematically impossible to happen randomly? Rational conclusion you are being followed, but why?

Please help me understand: { x|x is a an irrational number that is also rational }

Doing review for pre-calc and I'm honestly just a bit confused here. Google isn't being much help and my brain is just not working properly right now, plus math isn't my strong point. My question is...what on earth is this number?

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